influence of scale parameters of pearlite and heating rate on - csic
TRANSCRIPT
1
Kinetics and dilatometric behaviour of non-isothermal ferrite-to-
austenite transformation
F. G. Caballero, C. Capdevila and C. García de Andrés
Dr. F.G. Caballero and Dr. C. García de Andrés are in the Department of Physical
Metallurgy, Centro Nacional de Investigaciones Metalúrgicas (CENIM), CSIC,
Avda. Gregorio del Amo, 8, 28040 Madrid, Spain. Dr. C. Capdevila, is in the
Department of Materials Science and Metallurgy, University of Cambridge,
Pembroke Street, Cambridge CB2 3QZ, UK
2
A model that describes the ferrite-to-austenite transformation during continuous
heating in ARMCO iron and three very low carbon low manganese steels with a fully
ferrite initial microstructure is presented in this work. This model allows to calculate
the volume fraction of austenite and ferrite during transformation as a function of
temperature and thus to know the austenite formation kinetics under non-isothermal
conditions in full ferritic steels. Moreover, since dilatometric analysis is a technique
very often employed to study phase transformations in steels, a second model, that
describes the dilatometric behaviour of the steel and calculates the relative change in
length which occurs during the ferrite-to-austenite transformation, has been also
developed. Both kinetics and dilatometric models have been validated by comparison
of theoretical and experimental heating dilatometric curves. Predicted and
experimental results are in satisfactory agreement.
3
1. Introduction
The success of the current research on the modelling of microstructure and properties
in steel fabrication depends on the availability of reliable phase transformation
theory. A problem of particular importance in this respect is related to the kinetics of
austenite formation. Austenitisation is an inevitably occurring during the heat
treatment of the vast majority of commercial steels. Despite this consideration, there
has been little work carried out on the formation of austenite as compared to the large
amount of research on its decomposition. A quantitative theory dealing with the
nucleation and growth of austenite from a variety of initial microstructural conditions
is vital, especially in the fabrication of steel components by welding.1
Early work on austenitisation prior to 1940 was summarised in a paper by Roberts
and Mehl,2 which also reported a study of austenite formation from ferrite/pearlite
and ferrite/spheroidite aggregates establishing the nucleation and growth character of
the transformation. Posterior works indicated the importance of cementite
precipitates in ferrite in aiding nucleation of austenite,3,4 and considered austenite
growth controlled by cementite dissolution.3,5-7 These investigations give an
indication of the complexity of austenite formation in that the austenite is nucleating
and growing in a microstructure consisting of two phases which have different
degrees of stability.
In the eighties, the development of ferrite/martensite dual phase steels by partial
reaustenitisation revived the interest for the heating part of the heat treatment cycle.8-
13 In those works, the authors emphasised the importance of the microstructure
immediately before intercritical annealing. Roosz et al.14 determined the influence of
4
the initial microstructure on the nucleation rate and grain growth of austenite during
isothermal treatment of a eutectoid plain carbon steel.
Very little information is available about the austenite formation in steels subjected
to continuous heating.15 Recent work has allowed to model quantitatively the
transformation of an ambient temperature steel microstructure into austenite under
continuous heating condition.16 Lately, some investigators have adopted a different
approach to the problem using artificial neural network.17,18
In this work a model is presented to describe the ferrite-to-austenite transformation
during continuous heating in ARMCO iron and three very low carbon low
manganese steels with a fully ferrite initial microstructure. This model allows us to
calculate the volume fraction of austenite and ferrite during transformation as a
function of temperature and thus to know the austenite formation kinetics under non-
isothermal conditions in full ferritic steels. Since the ferrite-to-austenite
transformation is so rapid in these steels, then reliable measurements of the volume
fraction of austenite formed during the partial transformation cannot be obtained
experimentally. This renders the validation of the kinetics model through the
comparison of the experimental and calculated volume fraction of austenite formed
during the progress of the transformation to be impossible. However, dilatometric
analysis can be employed as an alternative technique to study phase transformation
kinetics in steels. In this sense, the relative change in length which occurs during the
ferrite-to-austenite transformation will be calculated as a function of temperature.
Both kinetics and dilatometric calculations will be validated by comparing
theoretical and experimental heating dilatometric curves.
5
2. Experimental Procedure
The chemical composition of the studied steels is presented in Table 1. All the steels
have a full ferrite microstructure (Fig. 1). Specimens were ground and polished using
standardised techniques for metallographic examination. Nital - 2pct etching solution
was used to reveal the ferrite microstructure by optical microscopy. The ferrite grain
size was measured on micrographs. The average ferrite grain diameter (D) (Table 2)
was estimated by counting the number of grains intercepted by one or more straight
lines long enough to yield, in total, at least fifty intercepts. The effects of a
moderately non-equiaxial structure was eliminated by counting the intersections of
lines in four or more orientations covering all the observation fields with an
approximately equal weight.19
To validate the dilatometric model and also, indirectly, the kinetics model for the
ferrite-to-austenite transformation, specimens of 3 mm thick and 12 mm long were
heated in a vacuum of 1 Pa at a constant rate of 0.05 K/s in an Adamel Lhomargy
DT1000 high-resolution dilatometer. The dimensional variations in the specimen are
transmitted via an amorphous silica pushrod. These variations are measured by a
linear variable differential transformer (LVDT) in a gas-tight enclosure enabling to
test under vacuum or in an inert atmosphere. The heating and cooling devices of this
dilatometer were also used to perform all the heat treatments. The DT1000
dilatometer is equipped with a radiation furnace for heating. The energy radiated by
two tungsten filament lamps is focused on the dilatometric specimen by means of a
bi-elliptical reflector. The temperature is measured with a 0.1 mm diameter Chromel-
6
Alumel (type K) thermocouple welded to the specimen. Cooling is carried out by
blowing a jet of helium gas directly onto the specimen surface. The helium flow rate
during cooling is controlled by a proportional servovalve. The high efficiency of heat
transmission and the very low thermal inertia of the system ensure that the heating
and cooling rates ranging from 0.003 K/s to 200 K/s remain constant.
3. Results and Discussion
3.1. MODELLING OF NON-ISOTHERMAL AUSTENITE FORMATION
KINETICS
Formation of austenite from ferrite is well established to be a nucleation and growth
process. Speich and Szirmae estimated the maximum ferrite/austenite interface
velocity as 0.016 m/s for a 200 µm ferrite grain diameter.20 This is a very high
velocity but still much less than that reported for diffusionless transformations, about
103 m/s.21
The possible nucleation sites for austenite in pure iron are either in the matrix, at
grain boundary faces, at grain boundary edges, or at grain corners.22 According to
Speich and Szirmae's work, matrix nucleation was not detected. All the austenite
nucleated at α/α grain boundaries, but grain boundary edges were favoured over
grain boundary faces as nucleation sites. Nucleation at corner sites could not be ruled
out. However, it was difficult to obtain conclusive evidence for these sites by
7
examination of two-dimensional sections. This suggests that all nucleation sites
should be considered for the modelling.
For grain boundary nucleated phase transformations at sufficiently high nucleation
rates, the potential nucleation sites are exhausted early in the transformation, so the
reaction is further controlled by growth. This behaviour is called 'site saturation'.
Cahn22 quoted that site saturation occurs when at least one nucleus per grain is
formed at a time shorter than D/G, where D is the average ferrite grain diameter and
G is the growth rate of austenite into ferrite. Dirnfeld et al.23 calculated that, for an
average ferrite grain diameter of 14 µm, one nucleus per grain is formed in a time
which is approximately equal to 0.025D/G at temperatures between 1033 and 1093
K. Then, consideration of site saturation for the nucleation of austenite into ferrite
would be a reasonable approximation in the modelling.
The only available theoretical treatment of boundary migration in the absence of
diffusion is that derived from absolute reaction rate theory.24 Here, the austenite
growth is controlled by processes at the interface and the growth rate G is given by:
{ }TgkTH
kS
kTg
kTG
kTG act γαγα δνδν →→
∆
∆
−
∆
=∆
∆
−= expexpexp (1)
where δ is the boundary thickness, ν is the number of attempts to jump the boundary
activation barrier per unit time, k is the Boltzman constant, T is the absolute
temperature, ∆Gact is the free energy for the activated transfer atoms across the
ferrite/austenite interface, ∆S is the entropy of activation per atom, ∆H is the
enthalpy of activation per atom, and ∆gα→γ is the Gibbs free energy difference per
8
atom between the α and γ phases. The values of ∆H and ν are uncertain but are
generally assumed to be equal to the enthalpy of activation for grain boundary
diffusion25 and to kT/h (being h Planck constant), respectively. The value of ∆S is
also uncertain and may be negative or positive. If we consider that the maximum
ferrite/austenite interface velocity for a 200 µm ferrite grain diameter is 0.016 m/s at
1223 K,20 ∆gα→γ = 41.87×10-24 J per atom, δ = 5 Å, and ∆H=276.33×10-21 J per
atom, then
∆
kSexpν is equal to 1.65×1017 s-1.
Thus, the growth rate of austenite into ferrite can be calculated as follows:
{ }
×−∆
×=
−→
kTTg
kTG
219103.276exp103.82 γα
(2)
Figure 2 shows the Gibbs free energy change for the ferrite-to-austenite
transformation ∆gα→γ for all the studied steels. This energy has been obtained
according to the thermodynamic calculations proposed by Aaronson et al.26,27 and
Kaufman et al..28 In order to account for the effects of alloying elements into
calculation, Zener factorization of the free energy into magnetic and non-magnetic
components has been performed.29
Assuming that site saturation occurs and the reaction is controlled by growth, the
kinetics law obtained for the three different activated growth sites can be expressed
as follows:22,23
9
++−−=
32exp1 tKtKtKV cesγ (3)
where Vγ represents the formed austenite volume fraction, t is the time and Ks, Ke and
Kc are given by,
VCGK
VLGK
VSGK ces
32
342 ππ === (4)
where VS ,
VL and
VC respectively are the boundary area, the edge length, and the
grain corner number, all per unit volume, and G is the growth rate of austenite.
Assuming ferrite grains to be tetrakaidecahedra,22 VS ,
VL and
VC can be expressed in
terms of the average ferrite grain diameter D by:
32125.835.3DV
CDV
LDV
S=== (5)
and thus,
3
3
2
23.507.267.6
D
GKD
GKD
GK ces === (6)
where G is given by equation (2).
10
The difficulties in treating non-isothermal reactions are mainly due to the complex
variations of growth rate with temperature, described in equation (2). We can only
deal with the problem when the rate of transformation depends exclusively on the
state of the assembly and not on the thermal path by which the state is reached.24
Reactions of this type are called isokinetic. Avrami defined an isokinetic reaction by
the condition that the nucleation and growth rates are proportional to each other (i.e.
they have the same temperature variation). This leads to the concept of additivity and
Scheil's rule.30
Since Avrami's condition for an isokinetic reaction is not satisfied in the present case,
a general equation to describe the non-isothermal overall ferrite-to-austenite
transformation in ferritic steels was derived integrating the equation (3) over the
whole temperature range where the transformation takes place.16 In this sense, we
have taken logarithms in equation (3), which then was differentiated,
32
11ln tKtKtKV ces ++=
− γ (7)
dttKtKKV
dVV
d ces
++=
−=
−2
3211
1lnγ
γ
γ (8)
If we consider a constant rate, •
T , for the heating condition, time can be expressed as
follows:
11
••
∆==
T
TtT
dTdt (9)
substituting into equation (8) and integrating in [ ]γV,0 and [ ]TTs , intervals on the
left and on the right sides, respectively, it can be concluded that:
( )•
••∫∫
∆+
∆+=
− T
dT
T
TKT
TKKV
dV T
Tces
V
s
2
2
0
321
γ
γ
γ (10)
thus,
( ) ( ) ( ) dTTTGDT
TTGDT
GDT
VT
TSS
s
∫
−
+−
+=−−••
•
233
22
8.1504.537.61ln γ (11)
where Ts is the start temperature of the transformation or temperature at which
0=∆→γα
g (root of the function represented in Fig. 2).
Therefore, the volume fraction of austenite (Vγ) and ferrite (Vα) present in the
microstructure as a function of temperature can be calculated, as follows,
( ) ( )
−
+−
+−−= ∫•••
dTTTGDT
TTGDT
GDT
VT
TSS
s
233
22
8.1504.537.6exp1γ (12)
12
γα VV −= 1 (13)
3.2 CALCULATION OF RELATIVE CHANGE IN LENGTH AS A FUNCTION
OF TEMPERATURE
Assuming that the sample expands isotropically, the change of the sample length ∆L
referred to the initial length Lo at room temperature is related to volume change ∆V
and initial volume Vo at room temperature for small changes as follows:
∆LL
V VVo
o
o=
−3
(14)
Therefore, ∆LLo
can be calculated from the volumes of the unit cells and the volume
fractions of the different phases present at every temperature during continuous
heating:
−+=
∆3
333
2
22
31
o
o
a
aaVaV
LL
o α
αγγαα (15)
with
( )[ ]( )[ ]3001
3001
−+=
−+=
Taa
Taa
o
o
γγγ
ααα
β
β (16)
13
where γα ,V are the volume fraction of ferrite and austenite, respectively at any
transformation temperature. o
aα is the lattice parameter of ferrite at room
temperature, and aα is the lattice parameter of ferrite at any transformation
temperature. The ferrite lattice parameter was taken to be that of pure iron,
866.2=o
aα Å. o
aγ is the lattice parameter of austenite at room temperature as a
function of the chemical composition of the austenite, and aγ is the lattice parameter
of austenite at any transformation temperature. γαβ , are the linear thermal expansion
coefficients of ferrite and austenite, respectively, in K-1.
The factor 2 in the numerator of equation (15) are due to the fact that, the unit cell of
ferrite contains 2 iron atoms, whereas that of austenite has 4 atoms. The austenite and
ferrite volume fractions were calculated at every temperature using equations (12)
and (13), respectively. The dependence of the lattice parameter of austenite on
alloying elements was taken from Ridley et al.31 and Dyson and Holmes,32
0.0018V+0.0031Mo+0.0006Cr+0.0002Ni-0.00095Mn+0.033C+3.573=o
aγ (17)
where the chemical composition is measured in wt-% and o
aγ is in Å.
The values of the linear thermal expansion of ferrite and austenite considered in these
calculations were βα = × − −1244 10 5 1. K and βγ = × − −2 065 10 5 1. K .33
14
3.3. EXPERIMENTAL VALIDATION OF KINETICS AND DILATOMETRIC
CALCULATIONS
The dilatation curves calculated using equation (15) for ARMCO steel and three low
carbon low manganese steels (Table 1) with a fully ferrite initial microstructure and
heated at a rate of 0.05 K/s are shown in Figs 3 and 4, respectively, in comparison
with their corresponding experimental results.
In Fig. 5 the experimental and calculated results of start (Ts) and finish (Tf)
temperatures of ferrite-to-austenite transformation are compared. Ts is considered to
be the temperature at which the relative change in length of the steel deviates from a
linear relation with temperature during heating due to the formation of austenite; Tf
has been defined as the temperature at which the sample exhibits again a linear
thermal expansion relation once the ferrite-to-austenite transformation is completed.
Points lying on the line of unit slope show a perfect agreement between experimental
and calculated values.
The calculated curves shown in Figs 3 and 4 suggest that the ferrite-to-austenite
transformation takes place almost instantaneously (1 K). In contrast, the experiments
reveal that this transformation needs between 10 and 20 K to reach completion at a
heating rate of 0.05 K/s.
Additionally, Figs 3, 4 and 5 show that experimental Ts and Tf temperatures are
higher than those predicted for all the studied steels. Any difference between these
represents some kinetic hindrance to transformation. Fig. 5 shows that the ARMCO
steel transforms to austenite at temperatures which are similar to the predicted
temperatures. The addition of manganese clearly leads to much larger deviations
15
from calculated results. That may be explained by the fact that the presence of a
substitutional solute retards the transformation to austenite because it is necessary for
the solute to diffuse during transformation.17
In general, the calculated relative change in length was consistent with the measured
value at every temperature. The fact that both the modelled and the experimental
dilatometric curves run parallel is irrelevant as long as the adequate hermal
expansion coefficients are calculated adequately.16 The linear expansion coefficients
of ferrite and austenite from Takahashi33 are in a good agreement with those
measured values.
16
4. Conclusions
1. A mathematical model applying the Avrami equation has been successfully
developed to reproduce the kinetics of the ferrite-to-austenite transformation in
ferritic steels during continuous heating. Nucleation of austenite occurs at the α/α
grain boundaries. All possible nucleation sites at the grain boundaries have been
taken in consideration in the modelling assuming that no nucluation barrier
exists. Since ferrite/austenite boundary migrates in the absence of diffusion, the
growth of austenite has been considered to be controlled by processes at the
interface.
2. The relative change in length which occurs during the ferrite-to-austenite
transformation has been calculated as a function of temperature. Experimental
values of relative changes in length and linear expansion coefficients measured
by high resolution dilatometry are accurately reproduced by calculations.
3. Experimental validation of the kinetics model of the ferrite-to-austenite
transformation has been carried out by comparison between experimental and
theoretical heating dilatometric curves. Results show that experimental start and
finish temperatures of the transformation are higher than those predicted for all
the studied steels. Furthermore, the addition of manganese clearly leads to much
larger deviations from calculated results since the presence of a substitutional
solute retards the transformation to austenite.
17
5. Acknowledgements
The authors acknowledge financial support from Consejería de Educación y Cultura
de la Comunidad Autónoma de Madrid (CAM07N/0065/1998).
18
6. References
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1991, A131, 99-113.
19
16. C. GARCÍA DE ANDRÉS, F. G. CABALLERO, C. CAPDEVILA and H. K. D.
H. BHADESHIA: Scripta Mater., 1998, 39, 791-796.
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21. R. F. BUNSHAH and R. F. MEHL: Trans. TMS-AIME, 1953, 197, 1251-1258.
22. J. W. CAHN: Acta Metall. 1956, 4, 449-459.
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20
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21
Table 1 Chemical Composition (Mass %)
Steels C Mn Si N Al P Cr Ni
ARMCO 0.002 0.05 - 0.004 - 0.003 - -
C-0.25Mn 0.010 0.25 0.028 0.003 0.057 0.016 0.014 0.022
C-0.37Mn 0.010 0.37 0.028 0.002 0.069 0.016 0.016 0.022
C-0.50Mn 0.010 0.50 0.028 0.004 0.046 0.015 0.012 0.020
22
Table 2 Average Ferrite Grain Diameter
Steels D / µm
ARMCO 158±28
C-0.25Mn 21±3
C-0.37Mn 63±11
C-0.50Mn 17±1
23
1 Optical micrograph of a full ferrite microstructure. ARMCO steel
-100
0
100
200
300
400
500
970 1020 1070 1120 1170 1220 1270
TEMPERATURE / K
∆gα
−γ / J/
mol ARMCO
C-0.25Mn
C-0.37Mn
C-0.50Mn
2 Gibbs free energy change for α→γ transformation
24
0,000
0,002
0,004
0,006
0,008
0,010
0,012
0,014
0,016
1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270TEMPERATURE / K
REL
ATI
VE C
HA
NG
E IN
LEN
GTH
Experimental
Calculated
ARMCOD =158 µm
3 Calculated dilatation curve of ARMCO steel compared with the
experimental curve obtained at a heating rate of 0.05 K/s
25
0,000
0,002
0,004
0,006
0,008
0,010
0,012
0,014
0,016
1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270
C-0.25MnD =21 µm
0,000
0,002
0,004
0,006
0,008
0,010
0,012
0,014
0,016
1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270
REL
ATI
VE C
HA
NG
E IN
LEN
GTH
Experimental
Calculated
C-0.37MnD =63 µm
0,000
0,002
0,004
0,006
0,008
0,010
0,012
0,014
0,016
1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270TEMPERATURE / K
C-0.50MnD =17 µm
4 Calculated dilatation curve of three low carbon low manganese steels
compared with their corresponding experimental curves obtained at a
heating rate of 0.05 K/s